

A020497


Conjecturally, this is the minimal y such that n primes occur infinitely often among (x+1, ..., x+y), that is, pi(x+y)  pi(x) >= n for infinitely many x.


22



1, 3, 7, 9, 13, 17, 21, 27, 31, 33, 37, 43, 49, 51, 57, 61, 67, 71, 77, 81, 85, 91, 95, 101, 111, 115, 121, 127, 131, 137, 141, 147, 153, 157, 159, 163, 169, 177, 183, 187, 189, 197, 201, 211, 213, 217, 227, 237, 241, 247, 253, 255, 265, 271, 273, 279, 283, 289, 301, 305
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OFFSET

1,2


COMMENTS

a(n) purportedly gives the least k with A023193(k) = n; that is, this sequence should be the "least inverse" of A023193.
My web page extends the sequence to rho(305)=2047 and also gives a superdense occurrence at rho(592)=4333 when pi(4333)=591  the first known occurrence.  Thomas J Engelsma (tom(AT)opertech.com), Feb 16 2004
Tomás Oliveira e Silva (see link) has a table extending to n = 1000.
The minimal y such that there are n elements of {1, ..., y} with fewer than p distinct elements mod p for all prime p.  Charles R Greathouse IV, Jun 13 2013


REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, (2nd edition, Springer, 1994), Section A9.


LINKS

T. D. Noe, Table of n, a(n) for n = 1..672 (from Engelsma's data)
Thomas J. Engelsma, Permissible Patterns.
T. Forbes, Prime ktuplets
Daniel M. Gordon and Gene Rodemich, Dense admissible sets, Proceedings of ANTS III, LNCS 1423 (1998), pp. 216225.
D. Hensley and I. Richards, Primes in intervals, Acta Arith. 25 (1974), pp. 375391.
H. L. Montgomery and R. C. Vaughan, The large sieve, Mathematika 20 (1973), pp. 119134.
Tomás Oliveira e Silva, Admissible prime constellations
Ian Richards, On the incompatibility of two conjectures concerning primes; a discussion of the use of computers in attacking a theoretical problem, Bulletin of the American Mathematical Society 80:3 (1974), pp. 419438.
H. Smith, On a generalization of the prime pair problem, Math. Comp., 11 (1957) 249254.
Eric Weisstein's World of Mathematics, Prime kTuples Conjecture.


FORMULA

Prime(floor((n+1)/2)) <= a(n) < prime(n) for large n. See Hensley & Richards and Montgomery & Vaughan.  Charles R Greathouse IV, Jun 18 2013


CROSSREFS

Equals A008407 + 1. First differences give A047947.
Cf. A023193 (prime ktuplet conjectures), A066081 (weaker binary conjectures).
Sequence in context: A130568 A143803 A284894 * A023490 A032375 A089556
Adjacent sequences: A020494 A020495 A020496 * A020498 A020499 A020500


KEYWORD

nonn,nice


AUTHOR

Robert G. Wilson v, Christopher E. Thompson


EXTENSIONS

Corrected and extended by David W. Wilson


STATUS

approved



